All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$24.00$ in total. The next month, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$8.50$ each for students, and the group paid $$74.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 24}$ ${20x+8.5y = 74}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-14y = -96}$ ${20x+8.5y = 74}$ Add the top and bottom equations together. $ -5.5y = -22 $ $ y = \dfrac{-22}{-5.5}$ ${y = 4}$ Now that you know ${y = 4}$ , plug it back into $ {5x+3.5y = 24}$ to find $x$ ${5x + 3.5}{(4)}{= 24}$ $5x+14 = 24$ $5x = 10$ $x = \dfrac{10}{5}$ ${x = 2}$ You can also plug ${y = 4}$ into $ {20x+8.5y = 74}$ and get the same answer for $x$ ${20x + 8.5}{(4)}{= 74}$ ${x = 2}$ There were $2$ teachers and $4$ students on the field trips.